Section 15.3 Double integrals inpolar coordinates.

1 Section integrals in polar choose a point in the plane that is called thepole(or origin) and labeledO. Then we draw a ray (half-line)starting atOcalled thepolar axis. This axis is usually drown horizontally to the right and corresponds to thepositivex-axis in Cartesian any point in the plane, letrbe the distance fromOtoPand let be the angle (in radians) betweenthe polar axis and the lineOP. Then the pointPis represented by the ordered pair (r, ) andr, are use the convention that an angle is positive if measured in the counterclockwise direction from the polar axisand negative in the clockwise direction. IfP= 0, thenr= 0 and we agree that (0, ) represents the pole for any valueof.

2 In the Cartesian coordinate system every point has only one representation, but in the polar coordinate systemeach point has many representations. Since a complete counterclockwise rotation is given by an angle 2 , the pointrepresented by polar coordinates (r, ) is also represented by(r, + 2 n) and ( r, + (2n+ 1) ),wherenis any connection between polar and Cartesian coordinates isx=rcos y=rsin andr2=x2+y2tan =yxEquation for do not uniquely determine it whenxandyare given. Therefore, in converting from Cartesian topolar coordinates , it is not good enough just to findrand that satisfy equations. We must choose so that thepoint (r, ) lies in correct want to evaluate Rf(x, y)DA,whereRis apolar rectangleR={(r, )|a r b, }Change to polar coordinates in a Double polar coordinatesx=rcos dA= (x, y) (r, ) drd y=rsin 1where (x, y) (r, )= x r x y r y is theJacobianof the us find the , iffis continuous on a polar rectangleRgiven by 0 a r b, , where 0 2 , then Rf(x, y)dA= baf(rcos , rsin )r drd Example the integral RxydAwhereRis the region in the first quadrant that lies between the circlesx2+y2= 4 andx2+y2= continuous on a polar region of the formD={(r, )}

3 | , h1( ) r h2( )},then Rf(x, y)dA= h2( )h1( )f(rcos , rsin )r drd Example the integral DxdAwhereRis the region in the first quadrant that lies between the circlesx2+y2= 4 andx2+y2= a Double integral to find the area of the region inside the circler= 3 cos and outside thecardioidr= 1 + cos .3 Example polar coordinates to find the volume above the conez= x2+y2and below the spherex2+y2+z2=